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Documents de Travail du
halshs-01053480, version 1 - 31 Jul 2014
Centre d’Economie de la Sorbonne
How to manage a large and flexible nuclear set
in a deregulated electricity market from the point
of view of social welfare?
Pascal GOURDEL, Maria LYKIDI
2014.59
Maison des Sciences Économiques, 106-112 boulevard de L'Hôpital, 75647 Paris Cedex 13
http://centredeconomiesorbonne.univ-paris1.fr/
ISSN : 1955-611X
1
How to manage a large and flexible nuclear set in a deregulated
electricity market from the point of view of social welfare?
Pascal Gourdel
University of Paris 1 Panth´eon-Sorbonne,
Centre d’Economie de la Sorbonne, Paris School of Economics
Maison des Sciences Economiques, 75634 PARIS CEDEX 13, France
E-mail: Pascal.Gourdel@univ-paris1.fr
Maria Lykidi1
University of Paris 1 Panth´eon-Sorbonne,
Centre d’Economie de la Sorbonne, Paris School of Economics
Maison des Sciences Economiques, 75634 PARIS CEDEX 13, France
E-mail: maria.lykidi@orange.fr, Tel: 0033 (0) 664262445
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June 2014
Abstract
In the case of a large nuclear set (like the French set), nuclear production needs to be flexible to adjust to the predicted evolutions of the energy demand. Consequently, the dominant
position of nuclear in the national energy mix makes it responsible for the overall equilibrium
of the electricity system which is directly intertwined with social welfare. In a previous work,
we looked at producers own profits (short-term, inter-temporal) considering the equality between supply and demand. Here, we proceed with a full optimization of the social welfare
in an identical framework. Theoretically, the optimal production behaviour that maximizes
social welfare is characterized by a constant thermal production and a totally flexible nuclear
production given that the nuclear capacity is sufficient. Numerically, the significant amount
of nuclear capacities compared with thermal capacities in the French electricity market leads
to the same “paradoxical” production behaviour. Therefore, we conclude that social optimum
is ensured within our model by investing sufficiently in nuclear capacity. The optimal production scheduling determined by the social welfare maximization problem and the optimal
inter-temporal production problem are totally opposite.
Key words: Electric power, nuclear power plant, flexibility, nuclear fuel stock, thermal
generation, social welfare, total cost minimization.
JEL code numbers: C61, C63, D24, D41, L11.
1
Corresponding author
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1
Introduction
In France, nuclear power plants do not only operate at baseload but they also need to be
flexible2 to follow a part of the variable demand because of the very significant size of the
nuclear set (IEA (2008), Pouret and Nuttall (2007), Bruynooghe et al. (2010)). This permits
to ensure the overall equilibrium between supply and demand and therefore to avoid a potential
“blackout”. The monitoring report realized by the French energy regulator (CRE) in 2007 gives
an illustration of the operation of the French nuclear set (Regulatory Commission of Energy
(2007)). It illustrates the way that the nuclear generation set is managed in France, focusing
on the load-following ability which characterizes it. In the Ph.D. thesis of Lykidi (2014), we
determined the optimal production behaviour when producers maximize their profits (shortterm, inter-temporel) in a deregulated electricity market dominated by the nuclear energy.
Here, the optimization no longer considers only the benefits of the generators, it now takes
into account the “benefits” for the whole society: social welfare. The nuclear operators being
the main producers of electricity in their national electricity market may have to consider
constraints inherent in the public interest when they optimize the management of their nuclear
fleet even in a competitive market. Such a constraint already covers the equality between
supply and demand taking into account the threat of a “blackout”. Indeed, this constraint is
already considered in the determination of the optimal management of the nuclear production.
It covers both the short-term and the inter-temporal optimization of the nuclear production.
However, the production decisions of a very large nuclear set have considerable consequences
for the whole national electricity system and hence for the welfare of the society. This may
lead towards a complete optimization of the social welfare instead of the producers own profits.
The question of the management of the nuclear set in order to maximize social welfare could be
also motivated by the fact that the social acceptability3 of nuclear is never acquired definitively
and until now the economic arguments has not given a decisive answer whether or not nuclear
has to participate in the energy mix of a country. Therefore, through this new problem, we
determine the optimal production levels that maximize social welfare given that the thermal
capacity (e.g. coal, gas, etc.) as well as the nuclear capacity are exogenous.
In view of the periodical shutdowns of nuclear reactors for reloading their fuel, we introduce,
in the medium-term, the feature of the nuclear fuel “reservoir” - partly similar to an hydroreservoir. We aim to determine the management of the nuclear fuel reservoir in order to
maximize social welfare during the time horizon of our model which consists of a number of
campaigns. Each campaign corresponds to the period of production between two successive
moments of reloading. Its length is given by the maximum number of days during which a
nuclear unit produces until exhaustion of its fuel of reloading. It generally takes between 12
and 18 months (Source: EDF, CEA (2008)). During this period, nuclear producers have to set
their seasonal variation of reservoir’s nuclear fuel to satisfy the seasonal demand and maximize
their profits. From the modelling of the nuclear fuel reservoir result constraints intrinsic to the
inter-temporal management of the nuclear fuel stock during several campaigns. Moreover, we
take into account production constraints imposed by the flexible4 operation of nuclear reactors.
2
The new reactor EPR, which is an evolution of the pressurized water reactor (PWR), is an example of a
III+ generation nuclear reactor which is designed to accommodate load-following operation (AREVA (2005),
Goldberg and Rosner (2012)).
3
Germany is a typical example of a country which shuts down all its nuclear plants after the Fukushima
disaster because nuclear was no more acceptable to society.
4
A nuclear unit can vary its capacity level between the nominal capacity and the technical minimum. In the
case of an EPR, load follow enables planned variations in energy demand to be followed and can be activated
between 25% of nominal capacity (technical minimum) and 100% of nominal capacity (technical maximum)
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We also include in the set of production constraints, the constraints issued from the available
thermal generation capacity and the equality between supply and demand each month. The
maximization of social welfare could give some insights with respect to an alternative behaviour
of the nuclear production (optimal short-term production behaviour, optimal inter-temporal
production behaviour).
Section 2 provides a brief description of our modelling and of the assumptions made within
our model. A more detailed description can be found in the second chapter of the Ph.D. thesis
of Lykidi (2014) where we build our model and we look at the optimal short-term management
of flexible nuclear plants in a competitive electricity market as a case of competition with
reservoir. Therefore, the demand for electricity and the time horizon of the model and of the
campaign are themselves modelled in the same way. We also bring to mind the modelling
of the generating units and of the production costs is presented in the general case of N > 2
producers. In section 3, we proceed with the maximization of social welfare. First, we define the
set of feasible solutions and of its interior. Then, we look at the social welfare maximization
problem under the set of feasible solutions. In view of our hypothesis of perfectly inelastic
demand, we show that its resolution is not possible. This leads us to the resolution of an
equivalent optimization problem which consists of the minimization of the total production
cost. Theoretically, in the case that the production constraints are not saturated, we come
up with a novel property that entirely characterizes the optimal solutions of the social welfare
maximization problem. Numerically, we feed our model with some data in order to resolve
the social welfare maximization problem by using Scilab5 (Section 4). Simulation results of
the social welfare maximization problem are analyzed and then, they are compared with the
theoretical results and the results of the optimal inter-temporal production problem. In section
5, we conclude.
2
Model: Perfect competitive case
In this section, we present shortly our deterministic, dynamic model and the assumptions made
within it. We consider a perfect competitive market where each producer disposes an amount
of nuclear and thermal capacity. The price in the market is calculated each month according
to the merit order price rule6 . We aim to specify the actions that the social planner will take
to manage a large and flexible nuclear set in order to maximize the welfare of the society.
Nuclear plants operate at baseload and at semi-base load and thus, they respond to a part of
the variable demand. In particular, we take into account the medium-term horizon in which
nuclear follows the variations of seasonal demand. In the determination of the optimal production behaviour that maximizes social welfare we look at the management of the nuclear fuel
reservoir during several campaigns of production. A number of operational constraints regarding the inter-temporal management of the nuclear fuel stock (nuclear fuel storage constraints),
the flexible operation of nuclear units and the generation capacity (minimum/maximum pro(NEA/AEN (2011)).
5
Scilab is an open source, cross-platform numerical computational package and a high-level, numerically
oriented programming language. It can be used for numerical optimization, and modelling, simulation of
dynamical systems, statistical analysis etc.
6
The merit order is a way of ranking the available technologies of electricity generation in the same order as
their marginal costs of production. This ranking results in a combination of different generation technologies to
reach the level of demand at a minimum cost. The price in the market is therefore determined by the marginal
cost of the “last technology” used to equilibrate supply and demand (perfect competitive case). This technology
is also called marginal technology.
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duction constraints) as well as the continuous and never ending equality between supply and
demand (supply-demand equilibrium constraints) are considered in the maximization of the
social welfare.
For simplicity reasons and in the absence of access to detailed data the electricity importations/exportations are not taken into account within our model. Moreover, our work centers
only on the nuclear fuel storage and the optimal management of the nuclear fuel reservoir without considering the production coming from hydro units with possibility of storage (peaking7
power plants) because of the additional capacity and storage constraints which would increase
the complexity of the model. There exists an extensive literature that studies the optimal
management of hydro-reservoirs in mixed hydro-thermal competitive markets and where one
can see several modellings of the optimal production problem and notice the increased level of
difficulty from a theoretical and numerical point of view (Arellano (2004), Bushnell (2003)).
A more detailed justification of the overall assumptions of our model as well as mathematical
proofs of propositions which appear in this section can be found in the second chapter of the
Ph.D. thesis (Lykidi (2014)).
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2.1
Modelling the generating units
We study a competitive electricity market with N > 2 producers who manage both nuclear
and thermal generating units. A producer n = 1, · · · , N can operate with all types of nuclear
generating units. Moreover, each producer disposes of a certain amount of thermal capacity.
2.1.1
Concept of type
Among the nuclear generating units, we distinguish several essential intrinsic characteristics:
• available nuclear capacity,
• minimum capacity when in use,
• month of their fuel reloading.
In our model, the minimum capacity is proportional to the available capacity, and this proportion is the same for all “physical” nuclear reactors. Therefore, for each “physical” nuclear
reactor, we will focus on the month of fuel reloading, which permits us to define twelve “types”
of nuclear units. Each type indexed by j = 1, · · · , 12 corresponds to a different month of reloading of the nuclear unit. Then, a unit which belongs to the type of unit j = 1 (respectively
j = 2, · · · , j = 12) shuts down in the month of January (respectively February, · · · , December).
A nuclear plant8 may contain several “physical” nuclear reactors, which (for operational
reasons) do not reload on the same month. The characteristic “type” for the nuclear case
is not related to the plant but to the reactor. Each producer n = 1, · · · , N owns a precise
number of “physical” nuclear reactors that are grouped according to the month of reloading
(independently of the locations) in order to constitute units. Therefore, it can hold a certain
level of capacity from each type of nuclear unit.
The modelling regarding the thermal units is the same except that the minimum capacity is
equal to zero and that there is no month of reloading. There is a unique type of thermal units.
7
Peaking power plants are power plants that generally run only when there is a high demand, known as peak
demand, for electricity.
8
A nuclear power plant is a thermal power station in which the heat source arises from nuclear reactions.
A nuclear unit is the set that consists of two parts: the reactor which produces heat to boil water and make
steam and the electricity generation system in which one associates: the turbine and the generator. The steam
drives the turbine which turns the shaft of the generator to produce electricity (Source: SFEN).
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2.2
Modelling the production costs
We recollect the modelling of the production costs. The cost functions of both nuclear and
thermal production are common to all producers. The nuclear cost function is made of a fixed
part determined by the cost of investment, the fixed cost of exploitation and taxes and a variable
part which corresponds to the variable cost of exploitation and fuel cost. We assume that the
cost function of the nuclear production is affine and defined as
nuc nuc
nuc
Cn,j
(qnjt ) = an,j
nuc + bnuc qnjt .
The thermal cost function is also made of a fixed part which corresponds to the cost of
investment, the fixed cost of exploitation and taxes and a variable part covering the variable
cost of exploitation, the fuel cost, the cost of CO2 as well as the taxes on the gas fuel. We
assume that the thermal production has a quadratic cost function Cnth (.) which is the following:
th
th
th 2
Cnth (qnt
) = anth + bth qnt
+ cnth qnt
.
nuc
The nuclear and thermal cost functions are monotone increasing and convex functions of qnjt
th
and qnt
respectively. We choose a quadratic cost function and thus, an increasing marginal cost
for the thermal production because: (i) the thermal production results from different fossil fuel
generation technologies (e.g. coal, gas -combined cycle or not-, fuel oil), (ii) the high fixed costs
of thermal production need to be recovered, (iii) we want to keep our model simple by choosing
the simplest cost function for thermal (DGEMP & DIDEME (2003, 2008), MIT (2003, 2009),
Cour des Comptes (2012)).
2.3
Notations and constraints
• T : the time horizon of our model. Its length is chosen to be equal to 36 months9 beginning
by the month of January in order to obtain a sufficiently long time horizon to follow up
the evolution of the value of the optimal solutions and at the same time to be consistent
with the absence of the discount rate. The complexity of our model leads to compromise
refinement of the model and computational capacity by choosing a reasoning in months10
rather than weeks.
• Tcampaign : the time horizon of the campaign. A French nuclear producer has two main options
regarding the scheduling of fuel reloading (Source: EDF (2008), CEA (2008)):
• per third (1/3) of fuel reservoir (representing a reloading of reactor’s core per third of
its full capacity) that corresponds to 18 months of campaign and 396 days equivalent
to full capacity for a unit of 1300 MW,
• per quarter (1/4) of fuel reservoir (representing a reloading of reactor’s core per
quarter of its full capacity) that corresponds to 12 months of campaign and 258
days equivalent to full capacity for a unit of 1500 MW.
Both options11 of fuel reloading result from the operational schema of EDF (Electricit´e de
France) that is strategically chosen in order to optimize the allocation of the shutdowns of
nuclear reactors for reloading (EDF (2008, 2010)). So, the scheduling of fuel reloading is
entirely exogenous within our model (Bertel and Naudet (2004), CEA (2008)). Our goal
is to determine the optimal allocation of the nuclear fuel stored in the reservoir during the
9
The time horizon of the model is a multiplicative of twelve, being expressed in months. Therefore it could
be modified.
10
This reasoning is also met in articles which study the optimal management of hydro-reservoirs in mixed
hydro-thermal electricity systems (e.g. Arellano (2004), Bushnell (1998)).
11
In the case of a unit of 900 MW, the scheduling of fuel reloading is the following: (i) 1/3 of fuel reservoir
that corresponds to 18 months of campaign and 385 days equivalent to full capacity, (ii) 1/4 of fuel reservoir
that corresponds to 12 months of campaign and 280 days equivalent to full capacity.
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•
•
•
•
•
•
•
different campaigns of production for a reloading pattern provided by the French nuclear
operator via the model ORION. We retain a duration of campaign equivalent to 12 months
to get a cyclic model with a periodicity of one year. The one year period can be then
decomposed into 11 months being the period of production and 1 month corresponding to
the month of reloading of the fuel. We do not choose a campaign of 18 months because it
is not in accordance with the “good” seasonal allocation of shutdowns of the nuclear units
which consists of avoiding shutdowns in high demand periods (winter) and concentrating
them as much as possible in low demand periods (between May and September). In fact,
if the nuclear producer reloads fuel in summer when the demand is low the date of the
next reloading will be then in winter when the demand is high. The case of having both a
campaign of 12 and of 18 months is excluded in order to avoid complicate our model and
because the choice of normative duration of the campaign can not be changed for a given
nuclear reactor. The Nuclear Safety Authority (NSA)12 has to give the authorization for
any changes on the choice of duration of the campaign. Additional to that the optimal
allocation of the shutdowns of all 58 nuclear reactors for reloading is decided in advance
according to safety rules imposed by NSA.
Dt : the level of demand observed in month t = 1, · · · , T . The demand for electricity being
an exogenous variable is assumed perfectly inelastic mainly because in the short-term to
medium-term, we may consider that price variations can not be observed by consumers in
real time and consumers habits and prior investments in electrical devices can not change
immediately. If we include a price elasticity of demand in our model, it would have a
random value since there are no particular elements that enable to assess its value.
hyd
Qt : the hydro-production coming from the run-of-river13 hydro plants in month t =
1, · · · , T . We assume that the monthly run-of-river hydro production is constant over
the total time horizon of our model given: (i) the non-availability of the data with regard
to the seasonal variations of hydro production because of precipitation and snow melting, (ii) its low volatility caused by a relatively low standard deviation which leads to a
steady evolution of its monthly value near to the mean over a year. It is calculated by the
mean of the yearly production. In this way, we deduce a significant part of the base load
demand in order to have a more accurate picture of the demand served by the nuclear
and thermal units. The intermittency that determines the base load production of the
renewable energy plants makes our model more complex and additionally to this it is not
coherent with the deterministic character of our model which is why we do not consider
it.
nuc
: the level of the nuclear production during the month t = 1, · · · , T for the unit j of
qnjt
producer n.
n,j,nuc
Qmax : the maximum nuclear production that can be realized by the unit j of producer n
during a month. The nuclear capacity is an exogenous variable.
n,j,nuc
Qmin : the minimum nuclear production that can be realized by the unit j of producer n
during a month.
th
qnt : the level of the thermal production during the month t = 1, · · · , T for the producer n.
Qn,th
max : the maximum thermal production during a month for the producer n. It corresponds to
the nominal thermal capacity of producer n. A producer may use the thermal resources
12
The Nuclear Safety Authority (NSA) is tasked, on behalf of the state, with regulating nuclear safety in
order to protect workers, the public and the environment in France.
13
The run-of-river hydro plants have little or no capacity for energy storage, hence they can not co-ordinate
the output of electricity generation to match consumer demand. Consequently, they serve as baseload power
plants.
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to produce electricity until it reaches the level of demand of the corresponding month
respecting at the same time the constraint (2). The thermal capacity is an exogenous
variable.
• Qn,th
min : the minimum thermal production during a month for the producer n. There is no
minimum for thermal production Qn,th
min = 0.
The minimum
and maximum production constraints have the form:
n,j,nuc
nuc
Qmin 6 qnjt
6 Qn,j,nuc
max , if no reload during month t for unit j
(1)
nuc
qnjt = 0,
if unit j reloads during month t
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th
0 6 qnt
6 Qn,th
max
(2)
n,j
: the nuclear fuel stock of reloading available to the unit j of producer n. This stock will
• Sreload
be expressed thanks to the conversion between the quantity of energy and the corresponding number of days of operation at full capacity rather than expressing it in kilograms of
uranium or number of nuclear fuel rods. In our model, the number of days of operation
equivalent to full capacity is constant for all j, n and inferior than 11 months which permits and obliges at the same time to modulate the nuclear production. The nuclear fuel
n,j
is equal to the corresponding capacity of the units of type j of
stock of reloading Sreload
producer n (Capacityn,j,nuc ) multiplied by the number of hours equivalent to full capacity
during a campaign. More precisely, one has:
n,j
= 1× Capacityn,j,nuc × Number of days equivalent to full capacity×24
Sreload
which corresponds to the nuclear fuel stock of reloading over a campaign of production.
n,j
• St : the quantity of fuel stored in the nuclear reservoir and available to the unit j of producer
n at the beginning of the month t = 1, · · · , T . Evidently, we have Stn,j > 0. If t is the
month during which the producer n reloads the fuel of the reactor then, the stock at
n,j
.
the beginning of the following month (beginning of the campaign) is equal to Sreload
A producer has a quantity of nuclear fuel stock equal to zero at the end of a campaign
(beginning of the month of reloading) which means that it spends all its nuclear fuel stock
n,j
during the campaign. The reasons that lead us to this ascertainment
of reloading Sreload
mainly concern the implicit costs that result from not consuming the totality of the
nuclear fuel stock during a campaign. Moreover, a producer has to finish the period T
at least with the same quantity of nuclear fuel as the initial one (STn,j+1 > S1n,j ). The
consideration of this constraint is motivated by some arguments analytically exposed in
the second chapter (avoid to “over-consume” the nuclear fuel stock to reach the maximum
nuclear production level because of induced negative effects, assure that each new cycle
of simulations of 36 months starts with the same quantity of nuclear fuel (S1n,j )).
The nuclear fuel constraints for the nuclear unit j of producer n are defined as
follows:
j=1
P12 nuc
n,1
qn1t = Sreload
Pt=2
24
n,1
nuc
qn1t
= Sreload
Pt=14
T
n,1
nuc
t=26 qn1t = Sreload
j ∈ {2, · · · , 11}
Pj−1 nuc
n,j
t=1 qnjt = S1
Pj+12−1
n,j
nuc
t=j+1 qnjt = Sreload
Pj+2·12−1
n,j
nuc
t=j+12+1 qnjt = Sreload
PT
n,j
n,j
nuc
t=j+2·12+1 qnjt = Sreload − S1
j=12
P11 nuc
n,12
t=1 qn12t = Sreload
P23
n,12
nuc
qn12t
= Sreload
Pt=13
T −1 nuc
n,12
t=25 qn12t = Sreload
Table 1
We can see that the nuclear units of type {2, · · · , 11} have two additional constraints than
the nuclear units of type 1 and 12. This is because there exist, at the beginning and end of the
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game, campaigns that we will qualify as incomplete.
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2.4
Number of optimization variables and of optimization constraints
In our model, the total number of optimization variables is equal to N · (J · T + T ) = N · (12 ·
36 + 36) = N · 468. The number of constraints resulting from the equality between supply and
demand is T = 36. In addition, the number of nuclear fuel constraints is N · ((2 · K + 1) · (J −
2) + (2 · K) · 2) = N · ((2 · 3 + 1) · (12 − 2) + (2 · 3) · 2) = N · 82, where K represents the number
of campaigns within our model. Lastly, the number of minimum and maximum nuclear and
thermal production constraints is equal to N · (J · T + T ) = N · (12 · 36 + 36) = N · 468. Hence,
the total number of optimization constraints is equal to N · 550 + 36. Even in the case of a
unique producer (N = 1), the number of variables (468) and of optimization constraints (586)
are quite large which leads to computational difficulties. This is because, the level of difficulty
of the numerical program to compute a solution of an optimization problem is increasing with
respect to the size of the model (number of optimization variables, number of optimization
constraints).
In general, computational difficulties can result from: (i) the difficulty of the numerical
program in calculating a global optimum since it can stop running when it finds a first solution
which could be a local optimum of the optimization problem and not proceeding until it finds
a global optimum, (ii) the sensibility of calculations with regard to the initial point that one
chooses so that the program start running (different initial points can lead to different results),
(iii) the duration of calculations which is increasing with respect to the size of the model.
3
Maximization of social welfare
In this section, we study the maximization of social welfare under production and nuclear fuel
constraints as well as the supply-demand equilibrium constraints. Under the assumption that
demand is inelastic, we show that the social welfare maximization problem is equivalent to the
total cost minimization problem (same set of solutions) and therefore, we search for a solution
of the total cost minimization problem. Then, we give a property that characterizes the optimal
solutions when the production constraints are not binding what we call “interior” solutions.
Specifically, we prove that in the absence of binding productions constraints, the solutions of
the social welfare maximization problem (equivalently total cost minimization problem) are
completely determined by a constant thermal production.
3.1
A property of the “interior” solutions
The maximization of social welfare is an optimization problem which consists in the maximization of the total surplus. Total surplus results from the sum of consumer surplus (denoted by
SC) associated with a given level of production and the sum of producer surplus (denoted by
SP). Consumer surplus is the difference between the total amount that consumers are willing
and able to pay for electricity and the total amount that they actually do pay (electricity evaluated at the market price) (Renshaw (2005)). The surplus of producer is equal to its revenue
minus the variable costs or equivalently to the profit increased by the fixed costs (Varian (2006),
Renshaw (2005)). Without loss of generality, we may translate producer’s surplus by the fixed
costs.
The social welfare maximization problem is
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T
X
max
nuc )J
th T
((qjt
j=1 ,qt )t=1 ∈C
!
J
J
X
X
nuc
nuc th
SC(
qjt
+ qtth ) + SP (
qjt
, qt )
t=1
j=1
j=1
PN
P
nuc
nuc
th
We denote qjt
= n=1 qnjt
the aggregate nuclear production and qtth = N
n=1 qnt the aggregate
thermal production obtained both during month t. For example, the exogenous variable Qth
max
will now represent the aggregate maximum thermal production. In the “aggregate” case which
is a particular case, given the minimum/maximum nuclear production constraints ((1), (2)),
the set of feasible solutions C of the social welfare maximization problem is defined as
nuc
6 qjt
Qj,nuc
6 Qj,nuc
max , for all j, t
min
C = q ∈ M s.t.
0 6 qtth 6 Qth
for all t
max ,
nuc J
nuc J
The set M is defined by all the production vectors of the form q = ((qj1
)j=1 , · · · , (qjT
)j=1 ,
th
th
q1 , · · · , qT ) that respect the nuclear fuel constraints of table 1 as well as the supply-demand
equilibrium constraint (3) for all t
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J
X
hyd
th
q nuc
jt + q t = Dt − Qt .
(3)
j=1
The set M is affine and the set C is compact (closed and bounded) and convex.
We also define F as the relative interior14 of C (F = ri(C)). It has the following form
nuc
< qjt
< Qj,nuc
Qj,nuc
max , for all j, t
min
F = q ∈ M s.t.
0 < qtth < Qth
for all t
max ,
nuc
= 0 and thus, the strict inequality conNote that if unit j reloads during month t then qjt
nuc
straints that determine the nuclear production qjt in the set F are no more valid. Moreover,
we remark that the non-emptiness of the set F obviously depends on the values of the exogenous
j
hyd
j
j,nuc
th
variables (Qj,nuc
max , Qmin , Qmax , Sreload , S1 , Dt , Qt ) of the social welfare maximization problem.
Theoretically, the set F , which is a subset of C, is non-empty. Following some linear transformations in the actual form of the optimization constraints included in the set F and then using
the classical result of non-emptiness of the interior of unit simplex, we find a point contained
in F . Empirically, in subsection 4, we show that F is a non-empty set for our numerical data,
hence the above assumption complies with this particular data.
Consequently, we have to solve
max
nuc )J
th T
((qjt
j=1 ,qt )t=1 ∈C
Z
T
X
t=1
∞
pt
Dt (p∗t )dp∗t
"
#!
J
J
X
X
nuc
nuc
qjt
+ pt (
+ qtth ) −
Cjnuc (qjt
) − C th (qtth )
j=1
j=1
where Dt (.) is the demand function at time t. As we can see in the part of the objective function
which corresponds to consumer surplus (the indefinite integral of the demand function from the
price pt to the reservation price), the reservation price of a consumer, i.e. the maximum price
that a consumer is willing to pay for electricity, has an infinite value. This is because, in view
14
It is important to emphasize that, in the general case of n producers, the usual interior of C is empty since
M is an affine set that is not equal to Rn . Consequently, we focus on a classical generalization called relative
interior (for the notion of the relative interior of a set cf. for example Florenzano and Le Van (2001), Boyd and
Vandenberghe (2004), Pugh (2002)).
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10
of the assumption of inelastic demand, the demand function Dt (.) is constant. We also recall
that the price pt is given by the equality between supply and demand during the time t.
However, the formula defining consumer surplus does not make sense in the presence of inelastic demand (infinite value of surplus). In view of this remark, we focus on the variation of
consumer surplus. Nevertheless, the infinite value of consumers surplus leads to an indeterminate form of the variation of consumers surplus. In order to avoid this ambiguity, we define
explicitly the variation of consumer surplus when the price evolves from pt which is a level of
reference to pt by the following formula (∆)
Z pt
∆=−
Dt (p∗t )dp∗t
pt
This definition is coherent with the classical case (finite value of surplus).
We can now start the calculation of the integral to obtain
J
J
J
X
X
X
nuc
nuc
nuc
qjt
+ qtth ) − pt (
qjt
∆ = −Dt (pt − pt ) = pt Dt − pt Dt = pt (
+ qtth ) = Kt − pt (
qjt
+ qtth )
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j=1
j=1
j=1
P
nuc
+ qtth ). It is obviously a constant that depends on t.
where Kt = pt Dt = pt ( Jj=1 qjt
In view of these remarks, we will maximize the function
T
X
"
# "
#!
J
J
J
X
X
X
nuc
nuc
nuc
Kt − p t (
qjt
+ qtth ) + pt (
qjt
+ qtth ) −
Cjnuc (qjt
) − C th (qtth )
t=1
j=1
j=1
j=1
Accordingly, the social welfare maximization problem can be written as
max
nuc )J
th T
((qjt
j=1 ,qt )t=1 ∈C
T
X
Kt −
t=1
J
X
!
nuc
Cjnuc (qjt
) − C th (qtth )
(4)
j=1
or equivalently
min
nuc )J
th T
((qjt
j=1 ,qt )t=1 ∈C
T
J
X
X
t=1
!
nuc
Cjnuc (qjt
) + C th (qtth )
(5)
j=1
Therefore, we deduce that the social welfare maximization problem is equivalent to the
total cost minimization problem (5) (same set of solutions). If the solution of the social welfare
maximization problem belongs to the set F in which the production constraints are not binding,
we obtain a property given by the following proposition.
nuc J
Proposition 3.1 If there exists a solution ((ˆ
qjt
)j=1 , qˆtth )Tt=1 ∈ F such that the social welfare
is maximum on C then qˆ1th = qˆ2th = · · · = qˆTth .
Proof
A proof of this proposition is provided in the Ph.D. thesis on page 207 − 210 (Lykidi (2014)).
We remind that C is a compact set, thus the total cost minimization problem has solutions
on C. Nevertheless, it may not have solutions on the set F since it is not compact. Hence, the
existence of a solution of the problem (5) on F has the form of assumption in Proposition 3.1.
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3.1.1
Economic interpretation of the Lagrange multipliers of the social welfare
maximization problem (or the equivalent total cost minimization problem
(5))
In view of the proof of Proposition 3.1, we can interpret economically the Lagrange multipliers of
the equivalent to the social welfare maximization problem, total cost minimization problem (5).
We remind that µt is the Lagrange multiplier associated with the supply-demand equilibrium
k
constraint at each month t and λj is the Lagrange multiplier for the nuclear fuel constraint
of the unit j during the campaign k. Since qˆ ∈ F , the equation (3.20) (respectively (3.21))
observed on page 209 of the proof implies that the sign of the multiplier µ1 (respectively µ2 )
is strictly positive. By a symmetric argument, the Lagrange multiplier µt is strictly positive
(µt > 0) for all t. Hence, in view of equations (3.22) and (3.23) on page 209 of the proof,
1
k
the multiplier λ3 (respectively λj ) is strictly negative. Indeed, if an additional unit of nuclear
fuel became available for unit j during campaign k, the thermal production would decrease
which would lead to the augmentation of the nuclear production cost and the diminution of the
thermal production cost. However, the second effect that regards the decrease of the thermal
production cost is the most important. Consequently, the “additional” cost resulting from an
k
additional nuclear fuel unit and thus the value of the multiplier λj should be negative. The
k
k
multiplier λj indicates the “marginal value of nuclear fuel stock”, i.e. the additional cost |λj |
unit j would incur if the nuclear fuel stock decreased by one unit during the campaign k.
Let us now proceed with a proposition which shows that a constant thermal production is a
sufficient condition for optimality on C.
nuc J
Proposition 3.2 If ((ˆ
qjt
)j=1 , qˆtth )Tt=1 is a production vector belonging to C such that qˆ1th =
th
nuc J
th
qˆ2 = · · · = qˆT then ((ˆ
qjt )j=1 , qˆtth )Tt=1 is a solution of the social welfare maximization problem
(equivalently total cost minimization problem) on C.
Proof
A proof of this proposition is provided in the Ph.D. thesis on page 211 − 213 (Lykidi (2014)).
Remark 3.1 We can prove that the strict convexity of the total production cost function with
respect to the thermal production q th implies the unicity of solutions with respect to the thermal
component (Lykidi (2014)). Nevertheless, if we take into account the other variables which do
not influence the total production cost, the total production cost function is convex with regard
to q, which does not mean necessarily that the entire solution is unique.
3.1.2
Economic analysis of Proposition 3.1 and of Proposition 3.2
In view of Propositions 3.1 and 3.2 (pages 10, 11), we conclude that in the absence of binding
production constraints, a constant thermal production is a characteristic property of solutions
of the social welfare maximization problem. On the contrary, the nuclear production, which is
the first that is called to satisfy demand (according to the merit-order rule), is adjusted fully to
the seasonal variations of demand. This is a result of the behaviour of producers who use the
thermal capacity to produce the same quantity every month in order to meet demand. Note
that it means that the amplitude of demand has to be smaller than the nuclear capacity so
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12
that the equality between supply and demand is respected each month (more precisely, the
amplitude of demand
to be inferior than
of nuclear production that can be
PJ has
PJthe amplitude
j,nuc
j,nuc
T
T
realized [max(( j=1 Qmax (t))t=1 ) − min(( j=1 Qmin (t))t=1 )] given our numerical modelling.
Furthermore, this property signifies that thermal is the marginal technology even during seasons
of low demand. Consequently, prices are determined permanently by the marginal cost of fossil
fuel technologies and hence, they stay constant during the entire time horizon of the model.
4
Numerical modelling
In this section, we study the nuclear and thermal production levels as well as the storage
levels resulting from the social welfare maximization problem, within a simple numerical model
solved.
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4.1
Data
The data that we use within our numerical model is derived from the French electricity market
and it is collected by different entities and for different years because of the difficulty of collection. Specifically, consumption data is given by the French Transmission & System Operator
(named RTE) for the year 2007, the annual generation capacity of hydro (run-of-river) results
from the French nuclear operator (EDF) while the annual nuclear and thermal (coal and gas)
generation capacity comes from RTE for the year 2009, the nuclear fuel stock of reloading has
been provided by EDF for the same year. The fixed and variable costs of nuclear, coal and
gas generation are obtained by the official report “Reference Costs of Electricity Production”
issued by the ministry of industry (General Direction for Energy and Raw Materials (DGEMP)
& Directorate for Demand and Energy Markets (DIDEME)) and they are computed for the
year 2007. In view of the specific characteristics of the nuclear generation technology and of
its production cost, we also provide a short analysis regarding the impact of the discount rate
on the calculation of the nuclear cost, the economic consequences of a load-following mode of
operation of nuclear reactors on the nuclear cost as well as the main points of differentiation
between nuclear and thermal production cost (Bertel and Naudet (2004)) in the Ph.D. thesis
of Lykidi (2014). Then, we present in detail some specific data assumptions considered for our
numerical modelling with respect to: (i) the value of the exchange rate, of the discount rate,
of the cost of CO2 per ton and the price of coal and gas (ii) the computation of the coefficients
of the thermal production cost, (iii) the simulation of the capacity for each type of nuclear
unit and of the initial value of the nuclear fuel stock (S1j ), (iv) the calculation of the number of
days equivalent to full capacity, (v) the technical minimum and maximum for an EPR reactor
in order to determine the minimum and maximum nuclear production constraints. Finally,
we refer to a couple of economical results which can be concluded within our data base and
which we fully develop in the second chapter of the Ph.D. thesis of Lykidi: (i) the calculation
of the average nuclear cost here (37.25 euros per MWh) is near the scope of nuclear electricity
prices (37.5 - 38.8 euros per MWh) evaluated for the NOME15 law (Commission for Energy
Regulation (CRE) evaluate this range of prices in 2010 (before Fukushima accident in 2011 (Les
Echos (20/04/2011)) in order to recommend to EDF a just price for selling nuclear capacity
15
The “Nouvelle Organisation du March´e de l’Electricit´e” (NOME) law indicates the findings of the report of
the Commission Champsaur which suggests access to nuclear electricity of the French nuclear operator (EDF)
for all producers (Champsaur (2009)). Specifically, the NOME law forces EDF to sell at a competitive price to
alternative producers of electricity and gas (GDF Suez, E.ON, ENEL, Poweo, Direct Energy, etc.) a quarter of
its nuclear production until 2025. This price should include the total cost of the operating nuclear plants.
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to other producers (Le Monde (01/02/2011))), (ii) the total monthly thermal production cost
can be covered only in the case that alternative generation technologies with higher marginal
costs are called to satisfy demand (e.g. hydro-storage units, oil, etc.).
4.2
Simulation results
In our numerical model, we maximize social welfare (equivalently we minimize total production
cost) on the entire set of feasible solutions C.
Demand
80000
70000
60000
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50000
40000
30000
20000
10000
0
0
5
10
15
20
25
30
35
40
Time
Figure 1: Simulated demand (in MW)
Simulation results obtained by our numerical model show that nuclear follows entirely the
seasonal variations of demand16 by decreasing during summer and increasing during winter
while thermal has a constant level of production during the entire time horizon of our model.
The amount of nuclear capacity is such that the amplitude of demand does never exceed it
within our numerical model which means that in view of the constant level of thermal production, the equality between supply and demand is always respected with only nuclear operated
in load-following mode. Thus, thermal remains the marginal technology during the entire time
horizon of the model while the nuclear technology is never marginal, even during months of
low demand (see Figure 1, Figure 2, Figure 3, Figure 4). We also observe that both nuclear
16
The amounts of monthly demand Dt are obtained for the period January 2007 - December 2009. In
particular, the values of monthly demand during the period January 2007 - December 2007 come from our
historical data. Then, we reproduce these values by applying a positive rate of 1% per year on the monthly
demand for the years that follow (2008 and 2009) to take into account the increasing trend of demand from one
year to another. We did a rescaling on this data to take into account the diversity on the length of the months.
Note also that the monthly demand in 2007 results from the aggregation of the hourly demand found within
our historical data.
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Production
80000
70000
60000
50000
40000
30000
hydraulic production
nuclear production
20000
thermal production
10000
0
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0
5
10
15
20
25
30
35
40
Time
Figure 2: Simulated hydro(run-of-river)/nuclear/thermal production (in MW)
and thermal production do not saturate the minimum/maximum nuclear17 and thermal18 pro17
The maximum nuclear production during the month t given that some unit is inactive during this month
Nuclear production
70000
60000
50000
40000
30000
20000
10000
0
0
5
10
15
20
25
30
35
Time
Figure 3: Simulated nuclear production (in MW)
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15
Thermal Production
20000
18000
16000
14000
12000
10000
8000
6000
4000
2000
0
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0
5
10
15
20
25
30
35
40
Time
Figure 4: Simulated thermal production (in MW)
(month of reloading) is represented by the purple dotted line. This quantity is obviously below the nominal
capacity of the French nuclear set represented by the crossed purple line. The minimum nuclear production
Nuclear fuel stock
400000
350000
300000
250000
200000
150000
100000
50000
0
0
5
10
15
20
25
30
35
Time
Figure 5: Simulated nuclear fuel stock (in MW)
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16
duction constraints. Consequently, we verify (through a numerical test too) that the numerical
solution described in this section belongs to F , therefore our simulation results are in accordance with Proposition 3.1 presented on page 10 seeing that the thermal component of this
solution is constant. We also deduce that our numerical results are in line with Proposition 3.2
appeared on page 11 since a production vector which belongs to C and is characterized by a
constant thermal production constitutes a solution of the social welfare maximization problem
within our numerical model.
The essentially periodic evolution of the nuclear production results in “high” levels of nuclear
fuel stock during summer and “low” levels of nuclear fuel stock during winter because of the
seasonality that characterizes the variations of the nuclear production (high production during
winter − low production during summer). Consequently, we observe a periodic evolution for
the nuclear fuel stock as well as an oscillation around the “stock of reference”19 (see Figure 3,
Figure 5).
Price
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40
35
30
25
20
15
10
5
0
0
5
10
15
20
25
30
35
40
Time
Figure 6: Simulated price (in Euro/MWh)
In view of the merit order price rule, the price20 is determined by the thermal marginal cost
since thermal is permanently the marginal technology (see Figure 6). Additionally to this, we
can see that the price is constant during the entire period T because thermal is characterized
by a constant level of production which leads to a constant thermal marginal cost.
during the month t given that some unit is inactive during this month (month of reloading) is represented by
the purple line of asterisks.
18
The maximum thermal production during a month is represented by the white blue dotted line and corresponds to the nominal thermal capacity (including coal, gas, fuel, etc.) of the French set.
19
The “stock of reference” is represented by the blue dotted line which shows the value of stock at the
beginning, being also the value of stock at the end.
20
The red (respectively yellow) dotted line indicates the price level when nuclear (respectively thermal) is the
marginal technology.
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17
Cost
2000
1800
1600
1400
1200
1000
800
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0
5
10
15
20
25
30
35
40
Time
Figure 7: Total cost (in Euro (million))
The evolution of total cost is almost periodic and its value increases during winter when
we observe high levels of nuclear production and decreases during summer when we notice low
levels of nuclear production (see Figure 7). As expected, the values of total cost (respectively
total variable cost) resulting from the optimal short-term production problem and the optimal
inter-temporal production problem are higher than the optimal value of total cost (respectively
total variable cost) determined in this section (see Table 2). Equivalently, we can say that the
values of social welfare obtained in the optimal short-term production problem and the optimal
inter-temporal production problem are lower than the optimal value of social welfare given by
the resolution of the social welfare maximization problem. To end, we notice that the total cost
(respectively total variable cost) resulting from the optimal inter-temporal production problem
is relatively higher than the total cost (respectively total variable cost) coming from the optimal
short-term production problem. This implies that the value of social welfare is relatively lower
when we maximize the inter-temporal profit where we determine a global optimum of the
optimal production problem than when we maximize the current monthly profit where we look
at “local” solutions of the optimal production problem.
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Social welfare
maximization problem
Optimal inter-temporal
production problem
Optimal short-term
production problem
5.209 × 1010
5.261 × 1010
5.250 × 1010
1.023 × 1010
1.075 × 1010
1.064 × 1010
4.186 × 1010
4.186 × 1010
4.186 × 1010
Total cost
(in Euro)
Total
variable cost
(in Euro)
Total
fixed cost
(in Euro)
Table 2
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4.2.1
General remarks
In view of the remark 3.1, the solution is unique with respect to the thermal component but
considering the other variables which do not act on the total cost the whole solution is not
definitively unique.
Production
Production
80000
70000
60000
50000
40000
30000
hydraulic production
nuclear production
thermal production
20000
10000
0
0
10
20
30
40
50
60
70
80
90
Time
Time
Figure 8: Simulated hydro(run-of-river)/nuclear/thermal production (in MW) (T=84)
Let also us notice that despite the possible modifications of the length of the time horizon
T of the model, we do not obtain different production patterns. Therefore, we do not observe
any changes in the periodic evolution of the nuclear and thermal production in the case of a
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longer time horizon (e.g. for T = 84, see Figure 8).
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4.3
Social welfare maximization problem VS Optimal inter-temporal
production problem
In this section, we compare the theoretical and numerical results coming from the resolution of
the optimal inter-temporal production problem and the social welfare maximization problem.
In the first optimization problem, we determine the production levels that maximize the intertemporal profit of a producer under production and storage constraints (Lykidi (2014)). Specific
interest is given to constraints inherent to public interest and social welfare and in particular
those imposed by the equality between supply and demand at each month even in a competitive
market. These constraints are due to the important size of the french nuclear set which ensures
the majority of the domestic demand. The second optimization problem however, gives priority
to production decisions that fully maximize social welfare -and not only producers profit- under
identical constraints. Here, we take into account that the production decisions of a very large
nuclear set have a very significant effect on the equilibrium of the whole national electricity
system and consequently on the welfare of the society.
From a theoretical point of view, we find that the optimal production behaviour deduced
from the inter-temporal profit maximization problem is diametrically opposite to the optimal
production behaviour resulting from the social welfare maximization problem. More precisely,
when the minimum and maximum production constraints are not saturated, we observe that
the solutions of the optimal inter-temporal production problem are fully characterized by a
constant nuclear production while the solutions of the social welfare maximization problem are
entirely characterized by a constant thermal production. So, producers maximize their intertemporal profit by using nuclear as a baseload generation technology that produces always at
a steady rate leaving thermal to follow-up the seasonal variations of the demand. On the contrary, social welfare is maximized through a totally flexible management of nuclear production
which is adapted completely to demand’s seasonal variations resulting from a constant thermal production that covers the residual demand every month. Nevertheless, in both problems,
thermal is always the marginal technology, i.e. the generation technology that determines the
market price over the entire time horizon of the model.
Numerically, the comparison of the optimal production behaviour between the inter-temporal
profit maximization problem and the social welfare maximization problem does not provide
exactly the same conclusions since the optimal inter-temporal production behaviour is not
identical with the one resulted from the theoretical resolution of the problem. In our numerical model, the optimal production deduced from the “regularized” optimal inter-temporal
production problem is such that both nuclear and thermal production follow the variations
of demand in the medium-term. Optimal solutions of the problem that are characterized by
a constant nuclear production do not exist. On the contrary, the optimal solutions derived
from the numerical resolution of the social welfare maximization problem are not differentiated from those coming from the theoretical resolution of this problem since in both cases
thermal remains constant during the whole period T and hence nuclear follows entirely the
seasonal variations of demand. Consequently, in both numerical problems, nuclear units realize
a load-following operation. Thermal units adapt their production output to follow-up load in
the inter-temporal profit maximization problem while they produce the same quantity to meet
demand every month in the social welfare maximization problem. Regarding the marginality
duration of nuclear, we notice that it is significantly increased in the optimal inter-temporal
production problem. More precisely, nuclear is the marginal technology during periods of low
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demand, i.e. during the half of our model’s period T , while the maximization of social welfare
shows that thermal is the marginal technology during the total time horizon of our model.
At this point, it should be noted the paradoxical nature of both theoretical solutions which
consists of not modulating production (thermal and nuclear respectively) over the whole time
horizon T . It shows clearly the importance of the choice of capacity (missing here because
capacities are exogenous variables within our model) in order to avoid to build and maintain
unused capacity taking into consideration the fixed costs of generation technologies. From a
theoretical point of view, in the social welfare maximization problem, if the level of nuclear
capacity is sufficiently high, we obtain a constant thermal production since nuclear production
can adjust fully to the seasonal variations of demand without exceeding its minimum and
maximum production levels. In our numerical model, a decrease of 47% in nuclear capacity
(compensated by the thermal capacity) would impose to modulate the thermal production in
order to balance supply and demand and respect at the same time the production constraints.
Similarly, from a theoretical perspective, in the optimal inter-temporal production problem,
under the assumption that the amount of thermal capacity is sufficiently important, the thermal
units load-follow without violating the minimum and maximum production constraints, thus
leaving the nuclear units to produce at a constant rate. However, lower levels of thermal capacity
would result in a modulation of nuclear production so that the supply-demand equilibrium
constraints and the production constraints are satisfied. Indeed, in our numerical example, we
remark that the level of thermal capacity, being lower than the amplitude of demand, results in
a modulation of both nuclear and thermal production in order to satisfy the equality between
supply and demand.
5
Conclusion
In a time period during which nuclear as an electricity generation technology is questioned
significantly given the accident of Fukushima, we decided to determine the optimal production
levels that maximize social welfare in a country where the global equilibrium between supply
and demand depends totally on nuclear generation. The maximization of social “benefits” and
the maximization of producers own profits under the same constraints consisted two different
approaches that the social planner may look at and they led to very different optimal production
behaviours.
Initially, we showed that the problem of maximization of social welfare can not be resolved
within a framework of inelastic demand. To overcome this difficulty issued from this basic
assumption of our model, we focused on the resolution of the total cost minimization problem
by proving that it is equivalent to the problem of social welfare maximization (same set of
solutions).
On a theoretical level, we resolved the social welfare maximization problem in the absence of
binding production constraints which is a specific case (implying that capacities are significant).
In this case, we proved that the optimal solutions are fully characterized by a constant thermal
production. Therefore, under the assumption that the nuclear capacity is significant, the nuclear production follows fully the variations of demand permitting to the thermal production
to be constant. Obviously, the optimal production scheduling determined by the social welfare
maximization problem and the optimal inter-temporal production problem are completely opposite. Indeed, under the assumption that the production constraints are not saturated, social
welfare (respectively inter-temporal profit) is maximized when the thermal (respectively nuclear) units produce at a constant rate. Therefore, in the first problem, given that the nuclear
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capacity is sufficiently important, we have a totally flexible operation of nuclear plants and in
the second problem, if the thermal capacity is sufficiently high, thermal production can adapt
totally to demand’s variations so that customers requirements for electricity are always satisfied
in both cases. Clearly, thermal is the marginal technology during the entire time horizon T in
both optimal production problems.
From an empirical point of view, the thermal units realize the same amount of production
each month to cover the residual levels of demand, forcing nuclear to entirely follow the seasonal
variations of demand. Here, the amplitude of demand is not greater than the nuclear capacity and hence, given the constant level of thermal production, the load-following operation of
nuclear units does not lead to a violation of the minimum and maximum nuclear production
constraints in order to equilibrate supply and demand. In our numerical example, a significant
decrease of 47% of the nuclear capacity (compensated by the thermal capacity) would necessitate the simultaneous modulation of the thermal production in order to balance supply and
demand every month. This has shown the robustness of our results since they would change
only if an important reduction of nuclear capacities occurred.
The numerical resolution of the social welfare maximization problem showed a production
behaviour that is identical with the one resulting from its theoretical resolution. The fact that
nuclear capacities are very important with respect to thermal capacities in the French electricity
market leads to a “paradoxical” optimal production behaviour being that of a steady thermal
production which induces an entirely flexible nuclear production. For the same reason, in
the case of the inter-temporal profit maximization problem, we did not arrive numerically at
the same conclusions with those derived from its theoretical resolution. In France, the thermal
capacity is not enough in order to manage nuclear uniquely as a baseload generation technology
at the optimum. It has also to operate at semi-base load following a part of the variable
demand, therefore, the flexible operation of nuclear units is inevitable. Hence, numerically,
nuclear plants operate to follow load in both cases while thermal plants produce at a constant
rate in the first case (social welfare maximization problem) while they follow load in the second
case (inter-temporal profit maximization problem). The nuclear production is never marginal
when maximizing social welfare and it is marginal during low demand periods when maximizing
inter-temporal profit.
Our theoretical and numerical results showed that social optimum is ensured by investing
significantly in nuclear capacity which shows clearly the necessity of nuclear in the energy
mix of a country from the social welfare perspective within our model. On the contrary, in a
decentralized market, the optimum of producers is attained if we make significant investments
in thermal capacity (French case). Thus, we conclude that the amount of investments used
to add new capacities of a generation technology (e.g. nuclear) plays an important role in the
determination of the optimal production behaviour of the corresponding technology and of the
other generation technologies (e.g. thermal) that constitute the national energy mix.
Nonetheless, we do not really see such behaviours in the French electricity market. For
example, the report of CRE in 2007 indicates a duration of marginality of nuclear equivalent
to a total of 1 or 2 months per year. This time period is not so close to the duration of nuclear
marginality resulting from the theoretical and numerical results of the optimal inter-temporal
production problem (a total duration of 6 months per year). Moreover, the thermal units do not
always produce at a constant rate as we derived from the production behaviour that maximizes
social welfare. The fact that nuclear producers and particularly the French nuclear operator
(EDF) does not take into consideration assumptions and factors of our model which limit the
model to a certain degree but contribute to resolve the above optimization problems and find
an optimal solution is the main reason why we notice these divergences from the real world.
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From an economical point of view, all these different approaches suggested in our model and
the results which are deduced theoretically and numerically give insights in order to induce
conclusions for policy and industry and thus, it could be interesting for the system operator to
look at.
To conclude, in all cases, nuclear fuel modelled as a “reservoir” of energy follows the seasonal
variations of demand in a competitive electricity market where nuclear capacity exceeds thermal capacity to a significant degree. But even if nuclear power does not possess the greatest
part of the energy mix of a country (like France), it can be still operated at semi-base load
following a part of demand’s variations because, technically, modern nuclear reactors are capable of flexible operation. This could lead to a more significant use of nuclear in the electricity
production of a country and therefore a higher share of nuclear power as a percentage of its
national energy production especially since nuclear promotes: (i) reduction of CO2 emissions,
(ii) energy independence from fossil fuel generation technologies, (iii) large-scale deployment
of intermittent electricity sources (renewable energy), (iv) economic competitiveness of a country’s energy sector. All these factors play a very important role in the future of nuclear energy
worldwide.
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